The idea of the proof is to apply Lyapunov-Schmidt reduction, i.e., to determine a bifurcation equation whose solutions are in one-to-one correspondence with heteroclinic connections between equilibria (REF ) near one of the explicit solutions \(u_0\)  from (REF ) connecting the equilibria \(Z^0_-\)  and \(Z^\pi _-\) . In the present context this is known as Melnikov's method, see for example [1]}.
